Chapter 2. Thermodynamics, Statistical Mechanics, and Metropolis Algorithm (2.1~2.5) Minsu Lee Adaptive Cooperative Systems, Martin Beckerman

Contents 2.1 Introduction 2.2 Key concepts 2.3 The measure of uncertainty 2.4 The most probable distribution –2.4.1 Configurations and weight factors –2.4.2 Entropic forms 2.5 The method of Lagrange multipliers

Introduction In this chapter, we will –Establish the formal structure and accompanying technical vocabulary for studying adaptive cooperative systems –Use probabilistic (or stochastic) language –Examine concepts underlying the study of cooperative systems in the thermodynamic setting that gave them meaning –Delineate the correspondence between the probabilistic, universal formalism, and thermodynamics

Introduction Thermodynamics (therme: heat, dynamis: power) –Study of the conversion of energy into work and heat and its relation to macroscopic variables such as temperature and pressure. –Statistical thermodynamics: statistical predictions of the collective motion of particles from their microscopic behavior –Caloric theory (heat was regarded as a fluid substance)  Mechanistic theory (heat was correctly identified with molecular activity) History of thermodynamics –Daniel Bernoulli (1738) Modern billiard-ball model in his kinetic theory of gases (mechanistic view) –Sadi Carnot (1824) Mechanical model 1 st law of thermodynamics – Conservation of energy –Thermodynamic setting in terms of internal energy, work done by system, and heat absorbed by it 2 nd law of thermodynamics – Entropy –Existence of a non-decreasing parameter of state

Introduction History of thermodynamics –Max Planck ( ) Quantum theory –Answer problems posed by radiant heat phenomena –Formulate statistical mechanics by Boltzmann and Gibbs relating the macroscopic thermodynamic observations to microscopic, stochastic process –Clausius The increase in entropy of a system that is not thermally isolated is associated with the differential amount of heat absorbed –Shannon (1948) Generalization of the thermodynamic concept of entropy to that of a universal measure of uncertainty –Jaynes (1956) Entropy can be used for generating prob. distributions including those of the statistical mechanics of Boltzmann and Gibbs, in particular

Key Concepts Entropy (2.3) –Information-theoretic measure of missing information and uncertainty Maximum entropy distribution(2.4) –In the special case where the probabilities are known from measurements of the relative frequencies of a set of events, the maximum entropy distribution is the most probable one Method of Lagrange’s undetermined multipliers (2.5) –Deduce the maximally noncommittal form of the probability distribution for the vast majority of experimental instances, where the available data are limited

Key Concepts Statistical mechanics (2.6) –Formal structure of inferencing theory Concepts and formalism associated with statistical mechanics, and thermodynamics (2.7, 2.8) –Uniform prob. distribution, and the broad class of exponential forms that includes the Gibbs distributions, emerge as maximum entropy distributions subject to simple moment constraints –Micro-canonical and canonical (Gibbs) distribution Monte Carlo method (2.9) Markov chains (2.10) –Convergence theorem for Markov chains Metropolis sampling algorithm (2.11) –Designed to generate a markov chain in a manner that satisfies the requisite conditions for the convergence theorem to hold –For simulating a Gibbs distribution

The Measure of Uncertainty Suppose –X: a variable (values: x 1, x 2,..., x r ) –p i : probabilities for these values, where i=1,...,r –S: a function called entropy of the probability distribution a measure of the uncertainty associated with probability distribution the amount of missing information which, if provided, would allow us to infer with complete certainty the value for the variable X Determine the form of the entropy function based solely upon consistency conditions with respect to continuity, monotonicity, and composition

The Measure of Uncertainty Continuity –Uncertainty measure S, is a continuous function of the probabilities associated with each outcome Monotonicity –If the probabilities are all equal to one another, then the uncertainty A(r) is a monotonic increasing function of r When the possible outcomes are equally likely, the uncertainty increases as the number of possibilities grows Composition –If a choice can be decomposed into two successive choices, the original uncertainty measure should equal the weighted sum of the component uncertainties

The Measure of Uncertainty More generally, –Let us place each of the r prob. into one of s distinct groups, with prob. p 1 to p m1 placed into the first group, prob. p m1+1 to p m1+m2 put into the second group, and so on. –The prob. for observing one of the elements in the first group is q 1 =(p 1 +p p m1 ) the second group is q 2 =(p m p m1+m2 ) and so on –Composition rule assumes the general form q 1, q 2,... : weight factors (take into account that the uncertainties due to the additional groupings are encountered prob. q i )

The Measure of Uncertainty Deduce the form of uncertainty measure S –Represent probabilities as ratios of integers m i : –Subdivide each p i to the point where each grouping is generated by events with probability 1/M –Within each group there are m i elements –Assume that the m i are each equal to an integer m –Assume that we successively partition the elements into equally likely groups of equally likely elements, ( ∵ monotonicity) ( ∵ composition) or  (K: arbitrary positive constant)

The Measure of Uncertainty Entropy: measure of uncertainty as to the value of the variable X

The Most Probable Distribution Configurations and Weight Factors –N elements, place each element into one of M bins –n i : # of elements in the ith bin... –p i =n i /N : prob. of finding an element belonging to a particular bin –example situations Systems of N molecules distributed among i=1,...,M quantum states Messages containing N symbols chosen from an alphabet of M letters, the ith letter occurring n i times –A distribution of the N elements among the M cells constitutes a particular configuration of the system –# of ways a specific configuration can be realized multinomial coefficient

The Most Probable Distribution Configurations and Weight Factor –Ex.1) N=2, M=2 (1) n 1 =2, n 2 =0, (2) n 1 =1, n 2 =1, (3) n 1 =0, n 2 =2 Polynomial coefficients (weight): W = 1, 2, and 1, respectively –Ex. 2) M=2, N=6 & n 1 =2, n 2 =4  there are 15 configurations First of the two elements of cell one occurring first: 5 –122221, , , , First “1” occurring second: 4 –212221, , , Initial “1” third: 3 –221221, , Initial “1” fourth: 2 –222121, “1” fifth –222211

The Measure of Uncertainty Configurations and Weight Factors –Apply Stirling’s formula take the logarithm, & assume that the n i and N are large –Maximizing the entropy corresponds to maximizing the multiplicity or weight factor –Maximizing the entropy is equivalent to determining the set {n i }, which can be realized in the greatest number of ways –In those instances where the prob. can be interpreted as a frequency distribution, the maximum entropy distribution is the most probable one

The Most Probable Distribution Entropic Form –S = K ln W Essential in some formulations of statistical mechanics The relationship between S and ln W is represented as a proportionality by Boltzmann The full equality, with the constant K identified as Boltzmann’s constant –W = e S/K Rapid increase in the number of states with increasing entropy –Entropy is closely connected with probability –The entropy of a physical system in a definite state depends solely on the probability of the state

The Method of Lagrange Multipliers Limited data with constraints How to make reasonable inferences or construct hypotheses, from limited information?  Identification of measure of uncertainty  Determine the form of the prob. when the entropy is maximal subject to the constraints

The Method of Lagrange Multipliers Lagrange’s method of undetermined multipliers –Find the extremum of a function f(x,y) –For arbitrary independent variations in the x- and y-directions, the extremum will occur when –Find the extremum along a curve g(x,y)=C, where C is a constant –Combining two conditions –If λ denotes the common ratio (Lagrange multiplier) –Define a new generating function

The Method of Lagrange Multipliers Lagrange’s method of undetermined multipliers –Generalize the results (independent variables x 1,..., x n ) –Further extend to instances where we have k=1,...,m, m<n, constraint equation (fewer constraint equations than independent variables) –m Lagrange multipliers, one for each constraint equation –The conditions for an extremum becomes –The generating function becomes –Apply this procedure for finding the extremum of a function subject to constraints to derive the form of the maximally noncommittal probability distributions